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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"4036190c11bac0da5a81cbf73d7e325a99b436a0335b7a2a824a5a17b2f00123","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-ad5020a0c2e8a31d","role":"sequence","name":"basis_map_A013610 of A Jacobsthal triangle.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:       1;\n  1:       4,         3;\n  2:      24,        39,          9;\n  3:     128,       411,        225,         27;\n  4:     704,      3951,       3897,       1080,         81;\n  5:    3840,     36291,      57789,      27783,       4698,        243;\n  6:   20992,    325095,     788409,     586008,     167589,      19197,       729;\n  7:  114688,   2869755,   10232901,   11052423,    4756482,     904932,     75087,     2187;\n  8:  626688,  25109631,  128657025,  194346864,  118139877,   33198174,   4513968,   284310,    6561;\n  9: 3424256, 218512179, 1583775693, 3261240711, 2691800586, 1048111245, 207635238, 21209526, 1049760, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (8+6*x)*P(n-2,x) + 6*x*d/dx P(n-2,x) + (4+3*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 3*x;\nP[n_, x_] := P[n, x] = Expand[(8 + 6*x)*P[n - 2, x] + 6*x*D[P[n - 2, x], x] + (4 + 3*x)*P[n - 1, x] + 3*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (8 + 6*x)*v[i - 2] + 6*x*deriv(v[i - 2], x) + (4 + 3*x)*v[i - 1] + 3*x*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-ba196dcd01800741","steps":["basis A013610","basis A008277"],"edges":["e-900137e7780f9cba","e-4e7a799f74f666af"],"text":"A113953 \u2192 basis A013610 \u2192 basis A008277"},"headline":"A113953 \u2192 basis A013610 \u2192 basis A008277","chain":null}